Analytic functions of a complex variable: Cauchy-Riemann equations, conformal mappings, integration, Cauchy theorem, Taylor and Laurent series, residues, contour evaluation of definite integrals. Linear vector spaces: Inner products, linear operators, eigenvalue problems, functions of operators and matrices, Fourier transforms, Hilbert spaces, Sturm-Liouville theory, classical orthogonal polynomials, Fourier series, Bessel functions.